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Igumnov L. A., Metrikin V. S. On the Complex Dynamics in Simplest Vibrational Systems with Hereditary-Type Friction. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2018, vol. 18, iss. 4, pp. 433-446. DOI:


On the Complex Dynamics in Simplest Vibrational Systems with Hereditary-Type Friction


The dynamics of a number of vibrational systems, accounting for the forces of hereditary-type dry friction and a vibration limiter, are studied in the paper. The interaction between the vibration limiter and the vibrational system is assumed to obey Newton's hypothesis. A general mathematical model has been developed, which is a strongly nonlinear non-autonomous system with a variable structure. The dynamics of the mathematical model is studied numerically-analytically, using the mathematical apparatus of the point mapping method. The special feature of the studying approach is that a point map is not formed in a classical way (mapping Poincare surface into itself), but based on times of the relative rest of the vibrational system, which considerably simplified both the point mapping process and its detailed analysis. The presence of floating  boundaries  of plates of sliding motion required an original approach to point mapping and interpreting the results obtained. The developed investigation methodology and software product were used to study the phase-plane portrait of the mathematical model as a function of the characteristics of sliding friction forces and rest, as well as of the type and position of the limiter. Based on the character of the bifurcation diagrams variation, it was possible to find the main laws of the motion regimes alteration process (the occurrence of periodic motion regimes of arbitrary complexity and possible transition to chaos via the period-doubling process) with the changing parameters of the vibrational system (the amplitude and frequency of the periodic effect, forms of the functional relation describing the variation of the friction coefficient value of relative rest. The results obtained with and without accounting for a vibration limiter are also compared in the paper.


1. Ishlinsky A. Yu., Kragelsky I. V. O skachkah pri trenii [About horse racing]. Zhurnal tekhnicheskoj fiziki [Journal Technical Physics], 1944, vol. 14, iss. 4–5, pp. 276–282 (in Russian).
2. Kashchenevsky L. Ya. Stohasticheskie avtokolebaniya pri suhom trenii [Stochastic selfoscillations with dry friction]. Inzhenerno-fizicheskij zhurnal [Journal of Engineering Physics], 1984, vol. 47, no. 1, pp. 143–147 (in Russian).

3. Vetiukov M. M., Dobroslavsky S. V., Nagaev R. F. Avtokolebaniya v sisteme s harakteristikoj suhogo treniya nasledstvennogo tipa [Self-oscillations in the system with the characteristic of dry friction hereditary type]. Izv. AN SSSR. MTT, 1990, no. 1, pp. 23–28 (in Russian).
4. Metrikin V. S., Nagaev R. F., Stepanova V. V. Periodic and stochastic self-excited oscillations in a system with hereditary-type dry friction. Journal of Applied Mathematics and Mechanics, 1996, vol. 60, iss. 5, pp. 845–850. DOI:
5. Zaytsev M. V., Metrikin V. S. On the Theory of a Nonautonomous Dynamical System with Hereditary-Type Dry Friction. Vestnik of Lobachevsky State University of Nizhni Novgorod, 2012, no. 3, pt. 1, pp. 141–146 (in Russian).
6. Vetyukov M. M., Platovskih M. Yu. Frikcionnye avtokolebaniya v sisteme s odnoj idvumya stepenyami svobody [Friction self-oscillations in a system with one and twodegrees of freedom]. In: Sovremennye problemy mekhaniki i ee prepodavaniya v vuze [Contemporary problems of mechanics and its teaching in high school] : Proc. of AllRussia Scientific and Methodical Conf. St. Petersburg, A. F. Mozhaisky Military Space Academy, 2015, vol. 1, pp. 58–63 (in Russian).
7. Leine R. I., van Campen D. H., de Kraker A. Stick-Slip Vibrations Induced by Alternate Friction Models. Nonlinear Dynamics, 1998, vol. 16, iss. 1, pp. 41–54. DOI:
8. van de Vrande B. L., van Campen D. H., de Kraker A. An Approximate Analysis of Dry- Friction-Induced Stick-Slip Vibrations by a Smoothing Procedure. Nonlinear Dynamics, 1999, vol. 19, iss. 2, pp. 157–169. DOI:
9. Leine R. I., van Campen D. H. Discontinuous fold bifurcations in mechanical systems. Archive of Applied Mechanics, 2002, vol. 72, iss. 2–3, pp. 138–146. DOI:
10. Leine R. I., van Campenb D. H. Bifurcation phenomena in non-smooth dynamical systems. European Journal of Mechanics A/Solids, 2006, vol. 25, iss. 4, pp. 595–616. DOI:
11. Luo G. W., Lv X. H., Ma L. Periodic-impact motions and bifurcations in dynamics of a plastic impact oscillator with a frictional slider. European Journal of Mechanics A/Solids, 2008, vol. 27, iss. 6, pp. 1088–1107. DOI:
12. Utkin N. F., Kizhniaev Yu. I., Pluzhnikov S. K. Obrabotka glubokih otverstij [Deep hole machining]. Leningrad, Mashinostroenie, 1988. 269 p. (in Russian).
13. Kuznetsova T. I., Makarov B. G., Germans B. A. O gashenii avtokolebanij pri glubokom sverlenii [On the suppression of self-oscillations with deep drilling]. Kolebaniya i ustojchivost’ mekhanicheskih sistem [Oscillations and stability of mechanical systems], 1981, iss. 5, pp. 114–118 (in Russian).
14. Minkov M. L. Tekhnologiya izgotovleniya glubokih i tochnyh otverstij [Manufacturing technology for deep and precise holes]. Moscow, Mashinostroenie, 1965. 176 p. (in Russian).
15. Troitskiy N. D. Glubokoe sverlenie [Deep drilling]. Leningrad, Mashinostroenie, 1971. 176 p. (in Russian).
16. Potyagajlo M. V. Izgotovlenie glubokih i tochnyh otverstij [Making deep and precise holes]. Moscow, Leningrad, Mashgiz, 1947. 108 p. (in Russian).
17. Gorodetsky Yu. I. Sozdanie matematicheskih modelej slozhnyh avtokolebatel’nyh sistem v stankostroenii [Creation of mathematical models of complex auto-oscillatory systems in machine-tool construction]. Avtomatizaciya proektirovaniya [Automation design]. Ed. V. A. Trapeznikov. Moscow, Mashinostroenie, 1986, iss. 1, pp. 203–221. (in Russian).

18. Kudinov V. A. Dinamika stankov [Machine dynamics]. Moscow, Mashinostroenie, 1967. 359 p. (in Russian).
19. Bowden F. P, Leben L. The Nature of Sliding and the Analysis of Friction. Proceedings of the Royal Society, 1939, vol. 109, no. 938, pp. 1939. DOI:
20. Kragilsky I. V. Trenie i iznos [Friction and wear]. Moscow, Mashinostroenie, 1968. 480 p. (in Russian).
21. Feigin M. I. Vynuzhdennye kolebaniya sistem s razryvnymi nelinejnostyami [Forced oscillations of systems with discontinuous nonlinearities]. Moscow, Nauka, 1994. 285 p. (in Russian).
22. Neymark Yu. I. Metod tochechnyh otobrazhenij v teorii nelinejnyh kolebanij [The method of point mappings in the theory of nonlinear oscillations]. Moscow, Nauka, 1972. 471 p. (in Russian).

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